I've been seeing this game in a lot of "best of the year" lists but I'm wondering what makes it unique from the existing glut of turn-based grid battle games? I've already played a bunch of these, including:

Advance Wars series
Shining Force series
Final Fantasy Tactics Advance
Rebuild series
Bahamut Lagoon
(and a little bit of Fire Emblem which I bounced off)

What makes Into The Breach different? Will I get anything out of it that I didn't already get with the above games?

It's actually notably less complex than the games you have listed above. Closer to the classic Chess than the TBS TRPGs. Procedural generation and extreamly high replayabilty. Pain of thinking for a number of minutes before making your next move just like in Chess. Sutuations where you can do nothing -- loosing is fun.